Homogenization of the Stokes equations in a domain perforated by a thin truss

Abstract

We consider the homogenization problem of the stationary Stokes equations in a periodically perforated domain whose solid structure consists of a thin truss. That is, the width $a_ε$ of the truss tends to zero faster than the period size $ε$. We identify the different scaling regimes and derive the corresponding effective equations. In the critical regime, we obtain the Brinkman equations, while in the subcritical regime, we recover the unchanged Stokes equations. In a subrange of the supercritical regime, we derive Darcy's law as the effective equation. Furthermore, we explicitly compute the effective coefficients in the Darcy and Brinkman limits. Moreover, as a tool in the analysis, we establish the analogous homogenization result for a periodic forest of cylinders. Finally, we also consider the homogenization result for a fence of cylinders.